5,337
edits
Deep Learning: Difference between revisions
→Optimal Transport (Earth-Mover)
| Line 1,062: | Line 1,062: | ||
The dual of <math>\min_{G} W_1(P_Y, P_{\hat{Y}})</math> is <math>\min_{G} \max_{D} \left[ E[D(Y)] - E[D(\hat{Y})] \right]</math>. | The dual of <math>\min_{G} W_1(P_Y, P_{\hat{Y}})</math> is <math>\min_{G} \max_{D} \left[ E[D(Y)] - E[D(\hat{Y})] \right]</math>. | ||
The lipschitz of the discriminator can be enforced by weight clipping. | The lipschitz of the discriminator can be enforced by weight clipping. | ||
===How to evaluate GANs?=== | |||
;Inception Score | |||
Use a pre-trained network (Inception-v3) to map a generated image to its probabilities. | |||
<math>IS(G) = \exp \left( E_{x \sim P_{\hat{X}}} KL( p(y|x) \Vert p(y) ) \right)</math> | |||
Mutual Information interpretation: | |||
<math>\log(IS(G)) = I(G(Z);y) = H(y) - H(y|G(z))</math> | |||
* The first term <math>H(y)</math> represents diverse labels. | |||
* The second score represents high confidence. | |||
IS is misleading if it only generates one image per class. | |||
;FID Score | |||
Use a pre-trained network (Inception) to extract features from an intermediate layer. | |||
Then model the data distribution using multivariate Gaussian with mean <math>\mu</math> and covariance <math>\Sigma</math>. | |||
FID is Frechet Inception Distance. | |||
<math>FID(x, y) = \Vert \mu_{x} - \mu_{g} \Vert_2^2 + Tr(\Sigma_{x} + \Sigma_g - 2(\Sigma_x \Sigma_g)^{1/2})</math> | |||
===A Statistical Approach to GANs=== | |||
GANs do not have explicit probability models. | |||
This is in contrast to maximum-likelihood models like VAEs. | |||
GANs focus on minimizing distance between distributions. | |||
This yields high-quality samples but inability to sample likelihoods. | |||
VAEs maximize lower bound on likelihood. However, you get blurry samples. | |||
The key idea is to have an explicit model for the data: | |||
<math>f_{Y}(y|X=x) ~ exp(-l(y, G(x))/\lambda)</math> | |||
;Theorem (BHCF 2019) | |||
... | |||
Entropic GANs meat VAEs. | |||
===Distributionally Robust Wasserstein=== | |||
Robust Wasserstein: | |||
<math> | |||
\begin{aligned} | |||
\min_{P_{\tilde{X}}, P_{\tilde{Y}}} | |||
\end{aligned} | |||
</math> | |||
==Misc== | ==Misc== | ||